The accurate imaging of the underground structure is needed in order to find a large oil deposit in the field of oil and gas exploration. To obtain the image of underground structure, it is necessary to know the velocity information of the underground structure. Geophysicists seem to have found a way to reconstruct detailed subsurface velocity, since Professor Tarantola (1984) proposed the full waveform inversion. For decades, geophysicists have published thousands of articles (Luo and Schuster, 1991; Pratt et al., 1998, 1999; Ravaut et al., 2004; Sirgue and Pratt, 2004; Plessix et al., 2010; Xu et al., 2012; Brittan et al., 2013; Warner and Guasch, 2014; Alkhalifah and Choi (2014); Wu and Luo, 2014; Xie, 2015), in order to investigate how to obtain the information of underground structure by seismic waveform. A variety of full waveform inversion methods are proposed with the improvement of computing ability of computer cluster, in which one of the most influential methods is the multi-scale waveform inversion proposed by Bunks (1995). This method starts from the low frequency data and gradually moves towards the high frequency data. However, there are two problems for almost all the researchers, one of which is that there is lack of low frequency information in seismic data. In oil and gas exploration, the main frequency of seismic data is conventionally above 15 Hz, Baeten et al., (2013) demonstrated that the frequency contents below 2 Hz are required to properly recover the low-wave number structure.
However, most current acquisition systems do not provide these low-frequency data. The second difficult problem is that the difference between the seismic synthetic data corresponding to initial model and the observation seismic records is greater than half of the wavelength in the full waveform inversion, which results in that the inversion iteration cannot converge. This requires that the initial velocity model is very close to the real model.
In view of the above-mentioned, a new solution is needed to solve the above problems.